184_notes:resistivity

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184_notes:resistivity [2018/06/19 14:41] – [Conductivity] curdemma184_notes:resistivity [2021/02/27 04:07] (current) – [Making sense of $R$] bartonmo
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 Section 19.2 in Matter and Interactions (4th edition) Section 19.2 in Matter and Interactions (4th edition)
  
-[[184_notes:cap_charging|Next Page: Charging and Discharging Capacitors]]+/*[[184_notes:cap_charging|Next Page: Charging and Discharging Capacitors]]
  
-[[184_notes:r_energy|Previous page: Energy in Circuits]]+[[184_notes:r_energy|Previous page: Energy in Circuits]]*/
  
 ===== Resistors and Conductivity ===== ===== Resistors and Conductivity =====
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 ====Resistance==== ====Resistance====
-Before when we talked about resistors, we said that a resistor was a section or part of the circuit where the passage of electrons requires more energy (conventionally, that it resists the passage of electrons more than other parts of the circuit). We found there to be a larger electric field, a larger drift velocity, and a larger surface charge gradient over the resistor. **Resistance** is a way to quantify how much a resistor resists the passage of electrons based off the properties of the material (electron mobility and electron density) and the shape of the resistor. +[[184_notes:resistors|Before when we talked about resistors]], we said that a resistor was a section or part of the circuit where the passage of electrons requires more energy (conventionally, that it resists the passage of electrons more than other parts of the circuit). We found there to be a larger electric field, a larger drift velocity, and a larger surface charge gradient over the resistor. **Resistance** is a way to quantify how much a resistor resists the passage of electrons based off the properties of the material (electron mobility and electron density) and the shape of the resistor. 
  
-{{  184_notes:resistorshape.jpg?350}} +[{{  184_notes:resistor_shape.png?350|A piece of a resistor with a potential difference of $\Delta$ V from one end to the other, a length L, and a cross-sectional area of A.}}] 
-== Derivation of $R$ ==+ 
 +==== Derivation of $R$ ====
  
 For example, suppose we have a resistor that has a cross sectional area of $A$, a length $L$, and a potential difference of $\Delta V$ from one end to the other. If we //__assume a steady state current__//, then the electric field inside the resistor would be constant in magnitude and direction and would point along the length of the wire. We could use the relationship between electric potential and electric field then to write: For example, suppose we have a resistor that has a cross sectional area of $A$, a length $L$, and a potential difference of $\Delta V$ from one end to the other. If we //__assume a steady state current__//, then the electric field inside the resistor would be constant in magnitude and direction and would point along the length of the wire. We could use the relationship between electric potential and electric field then to write:
 $$\Delta V =- \int_i^f \vec{E} \cdot \vec{dl}$$ $$\Delta V =- \int_i^f \vec{E} \cdot \vec{dl}$$
  
-{{184_notes:resistorefielddl.jpg?300  }}+[{{  184_notes:resistor_efield_dl.png?300|Electric field direction in a resistor (shown by the red arrow) and the dl vector shown by the blue arrow.}}]
  
 Because $\vec{E}$ would point along the length of the wire, we would want to integrate along the length of the wire, which would mean that $\vec{E}$ and $\vec{dl}$ would be parallel. This simplifies the dot product to just a multiplication, leaving: Because $\vec{E}$ would point along the length of the wire, we would want to integrate along the length of the wire, which would mean that $\vec{E}$ and $\vec{dl}$ would be parallel. This simplifies the dot product to just a multiplication, leaving:
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 $$R =\frac{L}{\sigma A}$$ $$R =\frac{L}{\sigma A}$$
  
-== Making sense of $R$ ==+==== Making sense of $R$ ====
  
 Why does the bottom fraction make sense? A longer, thinner wire should be more resistive, so the geometric properties make sense (directly proportionally to $L$ and inversely proportional to $A$). A wire with higher conductivity should be less resistive, which also make sense (inversely proprtional to $\sigma$). Why does the bottom fraction make sense? A longer, thinner wire should be more resistive, so the geometric properties make sense (directly proportionally to $L$ and inversely proportional to $A$). A wire with higher conductivity should be less resistive, which also make sense (inversely proprtional to $\sigma$).
  
-Resistance has units of volts per amp, which is also called an ohm. An ohm is represented by a capital omega ($\Omega$). Sometimes you may see resistance rewritten in terms of **resistivity**($\rho$), which is simply the inverse of conductivity $\rho=1/\sigma$. So using resistivity, $R=\frac{\rho L}{A}$ - either version of resistance is fine. +**Resistance has units of volts per amp, which is also called an ohm.** An ohm is represented by a capital omega ($\Omega$). Sometimes you may see resistance rewritten in terms of **resistivity**($\rho$), which is simply the inverse of conductivity $\rho=1/\sigma$. So using resistivity, $R=\frac{\rho L}{A}$ - either version of resistance is fine. 
  
-=== Ohm's Model ===+==== Ohm's Model ====
  
 Perhaps equally as important, we can now relate the change in electric potential over a resistor to the resistance and current passing through the resistor. This model of resistance works well for low voltage and currents. This model is also often called "Ohm's Law": Perhaps equally as important, we can now relate the change in electric potential over a resistor to the resistance and current passing through the resistor. This model of resistance works well for low voltage and currents. This model is also often called "Ohm's Law":
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 ^    Micro    ^    Macro    ^   ^    Micro    ^    Macro    ^  
 |    $v_{avg}=uE$    |    $J = \sigma E$    |   |    $v_{avg}=uE$    |    $J = \sigma E$    |  
-|    $i=nAv_{avg}=nAUE$    |   $I=|q|i=\frac{\Delta V}{R}$    | +|    $i=nAv_{avg}=nAuE$    |   $I=|q|i=\frac{\Delta V}{R}$    | 
  
 ==== Examples ==== ==== Examples ====
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